Capillary surfaces arising in singular perturbation problems. (English) Zbl 1435.35030

Karakhanyan, Aram L.

Anal. PDE 13, No. 1, 171-200 (2020).

Summary: We prove some Bernstein-type theorems for a class of stationary points of the Alt-Caffarelli functional in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) arising as limits of the singular perturbation problem \[\begin{cases} \triangle u_\varepsilon(x)=\beta_\varepsilon(u_\varepsilon)& \text{in } B_1, \\ |u_\varepsilon|\le 1&\text{in } B_1, \ \end{cases}\] in the unit ball \(B_1\) as \(\varepsilon\to 0\). Here \(\beta_\varepsilon(t)=(1/\varepsilon)\beta(t/\varepsilon)\ge 0\), \( \beta\in C_0^\infty[0,1]\), \( \int_0^1\beta(t)\,dt=M> 0\), is an approximation of the Dirac measure and \(\varepsilon>0\). The limit functions \(u=\lim_{\varepsilon_j \to 0}u_{\varepsilon_j}\) of uniformly converging sequences \(\{u_{\varepsilon_j}\}\) solve a Bernoulli-type free boundary problem in some weak sense. Our approach has two novelties: First we develop a hybrid method for stratification of the free boundary \(\partial\{u_0> 0\}\) of blow-up solutions which combines some ideas and techniques of viscosity and variational theory. An important tool we use is a new monotonicity formula for the solutions \(u_\varepsilon\) based on a computation of J. Spruck. It implies that any blow-up \(u_0\) of \(u\) either vanishes identically or is a homogeneous function of degree 1, that is, \(u_0=rg(\sigma)\), \( \sigma\in \mathbb{S}^{N-1}\), in spherical coordinates \((r, \theta)\). In particular, this implies that in two dimensions the singular set is empty at the nondegenerate points, and in three dimensions the singular set of \(u_0\) is at most a singleton. Second, we show that the spherical part \(g\) is the support function (in Minkowski’s sense) of some capillary surface contained in the sphere of radius \(\sqrt{2M}\). In particular, we show that \(\nabla u_0:\mathbb{S}^2\to \mathbb{R}^3\) is an almost conformal and minimal immersion and the singular Alt-Caffarelli example corresponds to a piece of catenoid which is a unique ring-type stationary minimal surface determined by the support function \(g\).

MSC:

35B25	Singular perturbations in context of PDEs
35J61	Semilinear elliptic equations
49Q05	Minimal surfaces and optimization
35R35	Free boundary problems for PDEs

Keywords:

singular perturbation problem; free boundary regularity; capillary surfaces; global solutions

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